Browse > Article
http://dx.doi.org/10.4134/BKMS.2008.45.4.739

ON THE STABILITY OF A GENERALIZED CUBIC FUNCTIONAL EQUATION  

Koh, Hee-Jeong (DEPARTMENT OF MATHEMATICS EDUCATION COLLEGE OF EDUCATION DANKOOK UNIVERSITY)
Kang, Dong-Seung (DEPARTMENT OF MATHEMATICS EDUCATION COLLEGE OF EDUCATION DANKOOK UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.45, no.4, 2008 , pp. 739-748 More about this Journal
Abstract
In this paper, we obtain the general solution of a generalized cubic functional equation, the Hyers-Ulam-Rassias stability, and the stability by using the alternative fixed point for a generalized cubic functional equation $$4f(\sum_{j=1}^{n-1}\;x_j\;+\;mx_n)\;+\;4f(\sum_{j=1}^{n-1}\;x_j+mx_n\;x_j\;-\;mx_n}\;+\;m^2\sum_{j=1}^{n-1}\;(f(2x_j)\;=\;8f(\sum_{j=1}^{n-1}\;x_j)\;+\;4m^2{\sum_{j=1}^{n-1}}\;\(f(x_j+x_n)\;+\;f(x_j-x_n)\)$$ for a positive integer $m\;{\geq}\;1$.
Keywords
Hyers-Ulam-Rassias stability; cubic mapping;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64   DOI
2 K.-W. Jun and H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 867-878   DOI   ScienceOn
3 Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378   DOI   ScienceOn
4 J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309   DOI
5 Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434   DOI   ScienceOn
6 P. Gavrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436   DOI   ScienceOn
7 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224   DOI   ScienceOn
8 P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86   DOI
9 H. Y. Chu and D. S. Kang, On the stability of an n-dimensional cubic functional equation, J. Math. Anal. Appl. 325 (2007), no. 1, 595-607   DOI   ScienceOn
10 F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129   DOI
11 S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, 1960
12 D. H. Hyers, G. Isac, and Th. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publishing Company, Singapore, New jersey, London, 1997
13 D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153   DOI
14 G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228   DOI   ScienceOn
15 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300   DOI   ScienceOn
16 Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284   DOI   ScienceOn
17 Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130   DOI
18 Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292-293; 309
19 I. A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979